| L(s) = 1 | − 2.08·2-s − 3.67·4-s + 5·5-s + 15.0·7-s + 24.2·8-s − 10.4·10-s + 4.52·11-s − 46.0·13-s − 31.3·14-s − 21.1·16-s + 29.2·17-s − 69.5·19-s − 18.3·20-s − 9.41·22-s + 97.2·23-s + 25·25-s + 95.6·26-s − 55.2·28-s + 29·29-s + 28.8·31-s − 150.·32-s − 60.8·34-s + 75.2·35-s − 423.·37-s + 144.·38-s + 121.·40-s − 299.·41-s + ⋯ |
| L(s) = 1 | − 0.735·2-s − 0.459·4-s + 0.447·5-s + 0.812·7-s + 1.07·8-s − 0.328·10-s + 0.124·11-s − 0.981·13-s − 0.597·14-s − 0.330·16-s + 0.417·17-s − 0.840·19-s − 0.205·20-s − 0.0912·22-s + 0.881·23-s + 0.200·25-s + 0.721·26-s − 0.373·28-s + 0.185·29-s + 0.167·31-s − 0.830·32-s − 0.306·34-s + 0.363·35-s − 1.88·37-s + 0.618·38-s + 0.479·40-s − 1.14·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 2.08T + 8T^{2} \) |
| 7 | \( 1 - 15.0T + 343T^{2} \) |
| 11 | \( 1 - 4.52T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 97.2T + 1.21e4T^{2} \) |
| 31 | \( 1 - 28.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 423.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 7.16T + 1.03e5T^{2} \) |
| 53 | \( 1 - 602.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 139.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 866.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 823.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 201.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 89.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 589.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 926.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 840.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.783211126923982057508918650543, −8.302461239236179254800032115393, −7.39212622106990689326147152418, −6.61934696821673611261220392457, −5.16546570232164932698885064482, −4.89987680533214420906899861319, −3.64504041687329343166320754528, −2.19886293527253624781413073011, −1.28071623970104388664223390581, 0,
1.28071623970104388664223390581, 2.19886293527253624781413073011, 3.64504041687329343166320754528, 4.89987680533214420906899861319, 5.16546570232164932698885064482, 6.61934696821673611261220392457, 7.39212622106990689326147152418, 8.302461239236179254800032115393, 8.783211126923982057508918650543
